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In the present work, an introduction to the contact phenomena in multibody systems is made. The different existing approaches are described, together with their most distinctive features. Then, the term of coefficient of restitution is emphasized as a tool to characterize impact events and the algorithm for calculating the relative indentation between two convex-shaped bodies is developed. Subsequently, the main penalty contact models developed in the last decades are presented and developed, analysing their advantages and drawbacks, as well as their respective applications. Furthermore, some models with specific peculiarities that could be useful to the reader are included. The aim of this work is to provide a resource to the novice researcher in the field to facilitate the choice of the appropriate contact model for their work.

In anurans, the complexity of courtship calls may affect female mate choice. The current study suggests that nonlinear phenomena (NLP) components can contribute to increasing complexity in courtship calls and attracting female attention. The results of a recent study showed that calls of large odorous frog (Odorrana graminea) contained NLP components. However, whether the nonlinear components of courtship calls in O. graminea improve male attractiveness remains unknown. We hypothesized that female O. graminea would prefer males producing calls with a higher proportion of NLP components (P‐NLP‐C). To test this hypothesis, we recorded the advertisement calls of 28 males and confirmed that the P‐NLP‐C was significantly positively related to body size. We also measured the body size of natural amplectant males and non‐amplectant males in the field and found that amplectant males had larger body sizes than non‐amplectant males, and the results of two‐choice amplexus experiments similarly revealed a female preference for males with larger body sizes. Additionally, phonotaxis experiments also revealed that females preferred male calls with a high P‐NLP‐C. The results suggest that a higher P‐NLP‐C in calls can enhance male attractiveness, and the P‐NLP‐C may provide key information about male body conditions for female O. graminea. Our study provides a new insight for better understanding the role of NLP in anuran mate selection. The results suggest that a higher proportion of nonlinear phenomena components (P‐NLP‐C) in calls can enhance male attractiveness, and the P‐NLP‐C may provide key information about male body conditions for female Odorrana graminea. This study provides a new insight for better understanding the role of NLP in anuran mate selection.

Fenichel’s geometric singular perturbation theory and the blow-up method have been very successful in describing and explaining global non-linear phenomena in systems with multiple time-scales, such as relaxation oscillations and canards. Recently, the blow-up method has been extended to systems with flat, unbounded slow manifolds that lose normal hyperbolicity at infinity. Here, we show that transition between discrete and periodic movement captured by the Jirsa–Kelso excitator is a new example of such phenomena. We, first, derive equations of the Jirsa–Kelso excitator with explicit time scale separation and demonstrate existence of canards in the systems. Then, we combine the slow-fast analysis, blow-up method and projection onto the Poincaré sphere to understand the return mechanism of the periodic orbits in the singular case, ϵ = 0.

The Hirota–Maccari (HM) system is a fundamental model in wave propagation that has been widely utilized to investigate complex nonlinear phenomena in nonlinear optics, optical communications, and mathematical physics. The HM system sheds light on critical insights into soliton dynamics, wave interactions, and other nonlinear effects. Therefore, the main goal of this work is to compile new soliton structures to the HM system in hyperbolic, trigonometric, exponential, and rational forms, both single and combined version, by using three robust analytical approaches: the new extended rational sinh‐Gordon equation expansion method (ShGEEM), the ( G ′ / G , 1/ G )‐expansion method, and the new extended hyperbolic function method (EHFM). The solutions thus obtained are also tested for their validity and accuracy using Mathematica. In order to exhibit the physical properties of the soliton pulses, we depict 2D, 3D, and contour graphs by using suitable values of the parameters. These methods play a major role compared with other methods in the literature because new and more general solutions are obtained with additional free parameters. Remarkably, all the known solutions are special cases of our generalized solutions. An important feature of the proposed methods is their simplicity, robustness, and computational efficiency that can be widely extended to nonlinear partial differential equations in all scientific disciplines. The efficiency of this framework thus indicates at its application in addressing complicated problems in the future.

In this work, we aim to explore new exact traveling wave solutions for the reaction–diffusion equation, which describes complex nonlinear phenomena such as cell growth and chemical reactions in nature. Obtaining exact solutions to this equation is crucial for understanding aspects such as reaction activity and the diffusion coefficient. We solve the reaction–diffusion equation by using the Riccati equation as an auxiliary equation. By controlling the parameters in the Riccati equation, we obtained a large number of traveling wave solutions, many of which were not formerly recorded in other documents. Numerical simulations demonstrate the evolution of various traveling waves of the reaction–diffusion equation in time and space. These rich exact solutions and wave phenomena help to expand our knowledge of this equation.

Nonlinear optical processes, such as harmonic generation, are of great interest for various applications, e.g., microscopy, therapy, and frequency conversion. However, high-order harmonic conversion is typically much less efficient than low-order, due to the weak intrinsic response of the higher-order nonlinear processes. Here we report ultra-strong optical nonlinearities in monolayer MoS 2 (1L-MoS 2 ): the third harmonic is 30 times stronger than the second, and the fourth is comparable to the second. The third harmonic generation efficiency for 1L-MoS 2 is approximately three times higher than that for graphene, which was reported to have a large χ (3) . We explain this by calculating the nonlinear response functions of 1L-MoS 2 with a continuum-model Hamiltonian and quantum mechanical diagrammatic perturbation theory, highlighting the role of trigonal warping. A similar effect is expected in all other transition-metal dichalcogenides. Our results pave the way for efficient harmonic generation based on layered materials for applications such as microscopy and imaging. Harmonic generation is a nonlinear optical process occurring in a variety of materials; the higher orders generation is generally less efficient than lower orders. Here, the authors report that the third-harmonic is thirty times stronger than the second-harmonic in monolayer MoS2.

This review of scaling theories of magnetohydrodynamic (MHD) turbulence aims to put the developments of the last few years in the context of the canonical time line (from Kolmogorov to Iroshnikov–Kraichnan to Goldreich–Sridhar to Boldyrev). It is argued that Beresnyak's (valid) objection that Boldyrev's alignment theory, at least in its original form, violates the Reduced-MHD rescaling symmetry can be reconciled with alignment if the latter is understood as an intermittency effect. Boldyrev's scalings, a version of which is recovered in this interpretation, and the concept of dynamic alignment (equivalently, local 3D anisotropy) are thus an example of a physical theory of intermittency in a turbulent system. The emergence of aligned structures naturally brings into play reconnection physics and thus the theory of MHD turbulence becomes intertwined with the physics of tearing, current-sheet disruption and plasmoid formation. Recent work on these subjects by Loureiro, Mallet et al. is reviewed and it is argued that we may, as a result, finally have a reasonably complete picture of the MHD turbulent cascade (forced, balanced, and in the presence of a strong mean field) all the way to the dissipation scale. This picture appears to reconcile Beresnyak's advocacy of the Kolmogorov scaling of the dissipation cutoff (as $\\mathrm {Re}^{3/4}$) with Boldyrev's aligned cascade. It turns out also that these ideas open the door to some progress in understanding MHD turbulence without a mean field – MHD dynamo – whose saturated state is argued to be controlled by reconnection and to contain, at small scales, a tearing-mediated cascade similar to its strong-mean-field counterpart (this is a new result). On the margins of this core narrative, standard weak-MHD-turbulence theory is argued to require some adjustment – and a new scheme for such an adjustment is proposed – to take account of the determining part that a spontaneously emergent 2D condensate plays in mediating the Alfvén-wave cascade from a weakly interacting state to a strongly turbulent (critically balanced) one. This completes the picture of the MHD cascade at large scales. A number of outstanding issues are surveyed: imbalanced turbulence (for which a new, tentative theory is proposed), residual energy, MHD turbulence at subviscous scales, and decaying MHD turbulence (where there has been dramatic progress recently, and reconnection again turned out to feature prominently). Finally, it is argued that the natural direction of research is now away from the fluid MHD theory and into kinetic territory – and then, possibly, back again. The review lays no claim to objectivity or completeness, focusing on topics and views that the author finds most appealing at the present moment.

High-dimensional partial differential equations (PDEs) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonlinear due to the need to incorporate certain nonlinear phenomena in the model such as default risks, transaction costs, volatility uncertainty (Knightian uncertainty), or trading constraints in the model. Such high-dimensional fully nonlinear PDEs are exceedingly difficult to solve as the computational effort for standard approximation methods grows exponentially with the dimension. In this work, we propose a new method for solving high-dimensional fully nonlinear second-order PDEs. Our method can in particular be used to sample from high-dimensional nonlinear expectations. The method is based on (1) a connection between fully nonlinear second-order PDEs and second-order backward stochastic differential equations (2BSDEs), (2) a merged formulation of the PDE and the 2BSDE problem, (3) a temporal forward discretization of the 2BSDE and a spatial approximation via deep neural nets, and (4) a stochastic gradient descent-type optimization procedure. Numerical results obtained using TensorFlow in Python illustrate the efficiency and the accuracy of the method in the cases of a 100-dimensional Black–Scholes–Barenblatt equation, a 100-dimensional Hamilton–Jacobi–Bellman equation, and a nonlinear expectation of a 100-dimensional G -Brownian motion.

Collective effects leading to spatial, temporal or spatiotemporal pattern formation in complex nonlinear systems driven out of equilibrium cannot be described at the single-particle level and are therefore often called emergent phenomena. They are characterized by length scales exceeding the characteristic interaction length and by spontaneous symmetry breaking. Recent advances in integrated photonics have indicated that the study of emergent phenomena is possible in complex coupled nonlinear optical systems. Here we demonstrate that the out-of-equilibrium driving of a strongly coupled pair of photonic integrated Kerr microresonators (‘dimer’)—which, at the ‘single particle’ (that is, individual resonator) level, generate well-understood dissipative Kerr solitons—exhibits emergent nonlinear phenomena. By exploring the dimer phase diagram, we find regimes of soliton hopping, spontaneous symmetry breaking and periodically emerging (in)commensurate dispersive waves. These phenomena are not included in the single-particle description and are related to the parametric frequency conversion between the hybridized supermodes. Moreover, by electrically controlling the supermode hybridization, we achieve wide tunability of spectral interference patterns between the dimer solitons and dispersive waves. Our findings represent a step towards the study of emergent nonlinear phenomena in soliton networks and multimodal lattices.A pair of strongly coupled photonic microresonators shows nonlinear emergent behaviour, which can be understood by incorporating interactions in the theoretical description of nonlinear optical systems.